An efficient sampling scheme for the eigenvalues of dual wishart matrices

Despite the numerous results in the literature about the eigenvalue distributions of Wishart matrices, the existing closed-form probability density function (pdf) expressions do not allow for efficient sampling schemes from such densities. In this letter, we present a stochastic representation for the eigenvalues of 2×2 complex central uncorrelated Wishart matrices with an arbitrary number of degrees of freedom (referred to as dual Wishart matrices). The draws from the joint pdf of the eigenvalues are generated by means of a simple transformation of a chi-squared random variable and an independent beta random variable. Moreover, this stochastic representation allows a simple derivation, alternative to those already existing in the literature, of some eigenvalue function distributions such as the condition number or the ratio of the maximum eigenvalue to the trace of the matrix. The proposed sampling scheme may be of interest in wireless communications and multivariate statistical analysis, where Wishart matrices play a central role.The work of Ignacio Santamaria was supported in part by the Ministerio de Ciencia e Innovación and AEI/10.13039/501100011033, under Grant PID2019-104958RB-C43 (ADELE). The work of Víctor Elvira was supported in part by the Agence Nationale de la Recherche of France PISCES under Grant ANR-17-CE40-0031-01, and in part by the ARL/ARO under Grant W911NF-20-1-0126